Frederico V. Prudente, Antonio Riganelli and Antonio J. C. Varandas- Calculation of the Rovibrational Partition Function Using Classical Methods with Quantum Corrections

8/3/2019 Frederico V. Prudente, Antonio Riganelli and Antonio J. C. Varandas- Calculation of the Rovibrational Partition Functi

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Calculation of the Rovibrational Partition Function Using Classical Methods with Quantum

Corrections

Frederico V. Prudente, Antonio Riganelli, and Antonio J. C. Varandas*

Departamento de Qumica, UniVersidade de Coimbra, P-3049 Coimbra Codex, Portugal

ReceiVed: December 6, 2000; In Final Form: February 28, 2001

The rovibrational partition functions of diatomic molecules are calculated using a classical framework plusquantum, semiclassical, and semiempirical corrections. The most popular methods to calculate such correctionsare briefly reviewed and applied to the benchmark H2 molecule. A novel hybrid scheme is proposed andapplied to H2, HCl, and ArO. Each method is analyzed with a view to find an economical way to calculatesuch corrections for polyatomic systems.

1. Introduction

Accurate values of the rovibrational partition function (qvr)are frequently needed in chemistry and physics. They are

required to calculate the equilibrium properties of molecularsystems,1 and the rates of chemical reactions using transitionstate theory.2,3 In principle, qvr can be evaluated exactly inquantum statistical mechanics by carrying out the followingexplicit summation of Boltzmann factors

where ) 1/(kBT), kB is the Boltzmann constant, T is thetemperature, i is the rovibrational energy associated with statei, and gi is the corresponding degeneracy factor. However, thecalculation of highly excited states with spectroscopic accuracyis currently feasible only for systems with a few degrees offreedom,4-6 which limits the evaluation of the rovibrationalpartition function using eq 1 to small molecules and lowrotational states.7-10

In turn, qvr assumes in classical statistical mechanics theform1,11

with HCM(q, p) being the classical Hamiltonian, h the Planckconstant, n the number of degrees of freedom, q the generalizedcoordinate vector, and p its conjugate momenta. In turn, thesubscript B implies that the hypervolume of integration is

restricted to phase space regions corresponding to a bound statesituation, i.e., 0 e HCM(q, p) e De, where De is the dissociationenergy of the molecule12 (throughout this work we assume asreference energy the minimum of the potential energy surface).A major advantage of the classical approach is the appreciablysmaller computational cost in comparison to the quantum one,which allows a treatment of molecular systems with a large

number of degrees of freedom. Yet, it is well-known that qvrCM

overestimates qvrQM at low temperatures, while converging to

the latter at the high-temperature limit;1 for a recent discussionon the range of applicability of classical statistical mechanics

to calculate the vibrational partition function for triatomicsystems, see ref 13.

For molecular simulations it is essential to have an economical

recipe to introduce corrections into the partition functionsevaluated within the classical framework. The idea for introduc-ing such corrections comes from the early days of quantummechanics. Such corrections can be based on strictly quantum,semiclassical and semiempirical formulations. A well establishedstrictly quantum approach comes from the seminal work ofWigner14 and Kirkwood.15 They have shown that the quantumprobability function (and hence the quantum partition function)in phase space can be obtained through an expansion in powersofp ) h/2. The Wigner-Kirkwood (WK) expansion has beenextensively employed in molecular simulations of liquids,16-19

and to calculate the partition functions of hindered rotors20 and

diatomic molecules.21,22 Another procedure to correct qvrCM

from quantum mechanics is due to Green23 and Oppenheim and

Ross24 (GOR). Similar to the WK method, the GOR approachconsists of writing the quantum Hamiltonian within the phasespace formalism as an expansion in powers of p, which is thenused in eq 2; for applications of the GOR method in molecularsimulations, see ref 25.

Other approaches to correct the classical partition functioncome from path integral formulations,26,27 and are of semiclas-sical nature. One such a proposal due to Feynman and Hibbs 27

uses the Feynman path integral formulation of quantummechanics, and consists of approximating the integration of theenergy functional over all paths by using an effective potential.Since their proposal, several forms of the effective potentialhave been suggested and applied to various physical prob-lems,19,28-34 including the calculation of the molecular rovibra-

tional partition function.35 The second proposal comes fromMiller and co-workers36-39 and is based on an approximationof the Feynman path integrals by their classical counterpartwhich, are in turn approximated by using a semiclassicalpotential. Such an approach has been utilized in the context ofmolecular vibration-rotation dynamics.40

A simple semiempirical procedure to correct the classicalpartition function was suggested by Pitzer and Gwinn. 41 In thePitzer-Gwinn method, the quantum partition function is ap-proximated as the classical partition function scaled by the ratioof the quantum and classical partition functions for a reference

qvrQM

(T) )i

gi exp(-i) (1)

qvrCM

(T) )1

hnB exp {- H

CM(q,p)} dq dp (2)

5272 J. Phys. Chem. A 2001, 105, 5272-5279

10.1021/jp0043928 CCC: $20.00 2001 American Chemical SocietyPublished on Web 05/04/2001


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system in which the later assume analytical form. Usually thereference system is a generalized harmonic oscillator po-tential. This is a very useful approximation, with several vari-ants of it having been tested for realistic potential energysurfaces.22,28,35,42-45

The main goal of the present work is to study the efficiencyof the above methods and discuss their advantages and dis-advantages in molecular partition function calculations. We alsoseek to establish a simple procedure in which the computational

effort is relatively inexpensive and easy to generalize to mul-tidimensional systems. This employs an hybrid effective po-tential which is built from effective potentials reported byprevious authors. Our purpose follows a claim made in previouswork13,46 (hereafter referred to as papers I and II) to seek ageneral economical procedure for calculating accurate valuesof internal partition functions based on realistic potential energysurfaces.

For the evaluation of the phase space or configurationalintegrals which arise from the quantum, semiclassical, orsemiempirical corrections discussed here we employ the methoddescribed in papers I and II. The method is based on a MonteCarlo technique proposed by Barker to calculate the density ofstates in transition state theory.47 We have limited our study to

diatomic systems since exact quantum partition functions canbe easily computed to test the methodologies discussed in the

present work. However, both the approaches to correct qvrCM

and the Monte Carlo technique are given for the generaln-dimensional case.

As case studies, we consider H2, HCl, and ArO in their groundelectronic states. While the first two systems represent chemi-cally stable species with different well depths, the latter is aweakly bound van der Waals molecule. For H2, we have testednumerically all the methods reported above. Such a systematicstudy will help us to figure out a novel scheme to introducequantum corrections into the classical partition function. Thisnew approach is then tested on the HCl and ArO molecules.For H2, the calculations are limited to the vibrational partition

function to avoid the inclusion of nuclear spin degeneracy factorswhich arise when the rotational motion is considered. Moreoveras pointed out by Taubmann et al.,22 an exact calculation of thevibrational partition function for H2 is a rather challenging taskdue to the large anharmonicity of the associated potential energycurve. On the other hand, the breakdown of the classical pictureis more serious for vibrational than rotational motions, sincerotational quanta are much smaller than vibrational ones.However, for HCl, and ArO, we report calculations of the fullrovibrational partition function.

The paper is organized as follows. In section 2, we summarizethe technical details. This includes an outline of the Monte Carlotechnique used in the classical calculations, a description of thepotential energy curves, and the details referring to the calcula-

tion of the rovibrational energy levels. Section 3 reviews thevarious methods to include corrections in the classical vibrationalpartition function: the Wigner-Kirkwood (WK) expansion,Green-Oppenheim-Ross (GOR) expansion, linear approxima-tion of the classical path (LCP) approach, quadratic Feynman-Hibbs (QFH) approximation of Feynman path integral, and thePitzer-Gwinn (PG) approximation. A novel hybrid LCP/QFHmethod is also reported in the present work is reported in section4. Some conclusions are in section 5.

2. Technical Details

2.1. Monte Carlo Procedure. For all calculations reportedin sections 3 and 4, it will be necessary to solve a multidimen-

sional integral of the following general form:

where {xi} are variables of the integrand function f, and B standsfor an appropriate volume of integration in (k-1)-dimensionsdefined by

where B is a scalar and F a k-dimensional function. An exampleis the classical partition function, for which we have em-ployed13,46 an efficient Monte Carlo scheme to evaluate it forpolyatomic systems. This method is an adaptation of Barkersalgorithm,47 which is based on stratified (guided) samplingprocedures. The basic idea is to choose a sampling domainwhich coincides, as much as possible, with the integrationdomain. Thus, the variables will not be sampled independently.Instead, one establishes a hierarchy into the sampling proce-dure such that the efficiency of the Monte Carlo method ismaximized. As a result, the sampled points form a normalizedbut nonuniform distribution, requiring the use of weightingfactors.

The algorithm may be summarized in the following steps:1. Find x ) (x1, ..., xk) which defines the minimum of F (x1,

..., xk).

2. Find the interval (x1min, x1

max) for variable x1 with all othervariables fixed at the values obtained in step 1, according to

3. Sample randomly x1 within this range to obtain x1S,

according to

where is a random number between 0 and 1.4. Starting with i ) 2, find new values for the remaining (k

- i + 1) variables which define the minimum F (x1S, ..., xi-1

S , xi,..., xk)

5. Find the integration domain (ximin, xi

max) for xi with thefirst (i - 1) variables fixed at the sampled values and theother (k - i) variables at the values obtained in step 4, ac-cording to

6. Sample randomly xi inside this range to obtain xiS as in

step 3.7. Repeat steps 4-6 for i ) 3, ..., K until the value (xk

S) ofthe last variable is sampled. This will result in the selection ofa sampled point within the boundary surface B defined by eq4. Note that the range of the ith variable is conditioned withrespect to all values of the variables previously selected.

8. Calculate the weight factor for each sampled point g, xgS

) (x1S, ..., xk

S), according to

which represents the hypervolume associated to xgS.

I) ... B

f(x1, ..., xk) dx1...dxk (3)

B ) F (x1, ..., xk) (4)

F (x1min

, x2, ..., xk) ) F (x1max

, x2, ... , xk) ) B (5)

x1S)x1

min+ (x1

max- x1

min) (6)

F (x1S, ..., xi-1

S, xi

min, xi+1, ..., xk)

F (x1S

, ..., xi-1S

, ximax

, xi+1, ..., xk) ) B (7)

wg )i)1

k

(ximax

- ximin

) (8)

Rovibrational Partition Function J. Phys. Chem. A, Vol. 105, No. 21, 2001 5273


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9. Repeat steps 3-8 M times to calculate the integral in eq3 as

where fg ) f(xgS) is the function to be integrated and M is the

total number of sampled points. The standard deviation associ-ated with eq 9 is47

2.2. Potential Energy Curves. We summarize in this sectionthe potential energy curves used for H2, HCl, and ArO. Althoughmore sophisticated models48,49 are available for H2 and HCl,

we have adopted curves which have been extensively used byothers in calculations of the partition function.The H2 molecule can be a benchmark system for such an

application due to its strong anharmonicity, as discussed byTaubmann et al.22 It will be described by a Morse curve50

defined by

where De is the equilibrium dissociation energy, a parameter,and re the equilibrium interatomic distance; Table 1 gathers thenumerical values of the involved parameters. For this model,the vibrational spectrum is given analytically as51

under the condition

This leads to a total of 17 vibrational bound states for H 2For HCl, we employ the empirical Hulbert-Hirschfelder

curve52 defined by

where y is the reduced internuclear distance

and b and c are parameters defined by

The values ofA0, A1, and A2 depend on experimental constantsthrough the relations:

where re has the meaning assigned above, and De has beenobtained as

In turn, D0 is the spectroscopic dissociation energy, Be therotational constant, Re the vibrational-rotational coupling

constant, and e, exe, eye, eze are the anharmonicityconstants in the Dunham series expansion. Such experimentalparameters are tabulated in ref 53, and gathered in Table 2. Oneobtains De ) 0.16969Eh.

To represent ArO, we have utilized the Hartree-Fockapproximate correlation energy (HFACE) model54 proposed byVarandas and Silva for diatomic interactions of both chemicallystable and van der Waals molecules. It assumes the form

where

and

The values of the relevant parameters are given in Table 3, whileDe ) 2.7979937 10-4Eh.

2.3. Eigenvalue Calculations. For comparison with themethods based on the classical framework, we evaluate alsothe corresponding quantum sum-over-states in eq 1. For this,we must solve the one-dimensional Schrodinger equation

TABLE 1: Parameters for the Morse Potential of H2:65 AllQuantities in au

De ) 0.1744 ) 1.02764re ) 1.40201 ) 918.6446

I IM) 1

Mg)1

M

wgfg (9)

2 )1

M(M- 1)g)1

M

(wgfg- IM

)2

(10)

V(r) ) De{1 - exp [-(r- re)]}2

(11)

E )De[( + 12)]2

2

p

2

De- [( + 12)]

1/2 2

p

2

2

) 0, 1, ... (12)

5500 K: at T )9000 K the relative error is already 6%. As discussed in ref12, this arises due to the fact that eq 33 implicitly considersthe contribution of dissociative species.

3.4. The Quadratic Feynman-Hibbs Approximation. TheFeynman path integral formalism is one of the most appealingapproaches to evaluate the quantum mechanical partitionfunction.61 However, its application to large systems is com-putationally very demanding.62 In the context of quantum

corrections, a simple approximation which was proposed initiallyby Feynman and Hibbs27 consists of replacing the classicalpotential by an effective one. In this case, the partition functionassumes the form of qCM where the integration over momentais carried out explicitly (as in the LCP approach) and theclassical potential is replaced by an effective quantum one. Thesimplest is the quadratic Feynman-Hibbs27 potential obtainedby keeping only quadratic fluctuations around the classical paths.The QFH partition function assumes the form

TABLE 4: Vibrational Partition Functiona of H2 Calculated Using Various Approaches

T/K qvCM

qvWK

qvGOR

qvLCP

qvQFH

qvLCP/QFH

qvPG

qvQM

300 4.792(-2)b -8.416(-1) c 7.696(-3) 1.653(-9) 1.065(-6) 4.619(-5) 3.082(-5)1.8(-3)d 7.6(-2) c 2.2(-4) 1.8(-11) 8.4(-8) 1.8(-6)

500 7.978(-2) -4.474(-1) c 2.096(-2) 1.400(-4) 1.059(-3) 2.453(-3) 1.965(-3)2.3(-3) 3.5(-2) c 4.1(-4) 1.4(-6) 5.1(-5) 7.2(-5)

700 1.118(-1) -2.611(-1) c 3.991(-2) 4.266(-3) 1.002(-2) 1.348(-2) 1.166(-2)2.8(-3) 2.1(-2) c 6.2(-4) 3.9(-5) 3.6(-4) 3.3(-4)

1000 1.600(-1) -9.824(-2) c 7.706(-2) 3.209(-2) 4.382(-2) 4.856(-2) 4.444(-2)3.3(-3) 1.2(-2) c 9.4(-4) 2.7(-4) 1.1(-3) 1.0(-3)

1500 2.410(-1) 7.125(-2) c 1.545(-1) 1.181(-1) 1.292(-1) 1.338(-1) 1.277(-1)4.1(-3) 6.7(-3) c 1.5(-3) 8.9(-4) 2.5(-3) 2.3(-3)

2000 3.227(-1) 1.972(-1) c 2.415(-1) 2.165(-1) 2.244(-1) 2.283(-1) 2.219(-1)4.7(-3) 4.4(-3) c 2.0(-3) 1.5(-3) 3.5(-3) 3.3(-3)

3000 4.887(-1) 4.074(-1) 1.434(+2) 4.221(-1) 4.104(-1) 4.140(-1) 4.167(-1) 4.116(-1)5.7(-3) 2.4(-3) 3.0 2.9(-3) 2.6(-3) 5.0(-3) 4.9(-3)

4000 6.584(-1) 5.993(-1) 6.582(-1) 6.039(-1) 5.981(-1) 5.995(-1) 6.012(-1) 5.978(-1)6.6(-3) 1.7(-3) 5.5(-3) 3.7(-3) 3.5(-3) 6.1(-3) 6.0(-3)

5000 8.326(-1) 7.868(-1) 8.068(-1) 7.875(-1) 7.846(-1) 7.844(-1) 7.853(-1) 7.833(-1)7.4(-3) 1.4(-3) 6.5(-3) 4.4(-3) 4.2(-3) 7.1(-3) 7.0(-3)

6000 1.0122 9.752(-1) 9.860(-1) 9.767(-1) 9.755(-1) 9.716(-1) 9.718(-1) 9.709(-1)8.1(-3) 1.3(-3) 7.4(-3) 5.1(-3) 5.0(-3) 7.8(-3) 7.8(-3)

7000 1.1980 1.1672 1.1739 1.1782 1.1780 1.1630 1.1626 1.16278.7(-3) 1.4(-3) 8.2(-3) 5.7(-3) 5.6(-3) 8.5(-3) 8.5(-3)

8000 1.3908 1.3645 1.3689 1.4012 1.4016 1.3601 1.3591 1.35999.3(-3) 1.5(-3) 8.9(-3) 6.4(-3) 6.3(-3) 9.2(-3) 9.1(-3)

9000 1.5906 1.5676 1.5707 1.6563 1.6571 1.5634 1.5619 1.5630

9.9(-2) 1.7(-2) 9.5(-3) 7.0(-3) 6.9(-3) 9.8(-3) 9.7(-3)a See the text for nomenclature. b Given in parantheses is the power of 10 by which the numbers should be multiplied. c Unconverged results.

d Second entry for each temperature gives the standard Monte Carlo deviation.

Figure 1. Relative errors of the approximate vibrational partitionfunction of hydrogen molecule as a function of temperature: (b)

qvCM, (1) qv

WK, (9) qvGOR, (2) qv

LCP, (- ) qvQFH, (--) qv

PG, and

(s) qvLCP/QFH

(see the text for nomenclature).

qvrLCP

(T) ) ( 2p2)3n/2

dq

exp{-[V(q) +2

p2

24(V(q))

2]} (33)qvr

QFH(T) ) ( 2p2)

3n/2

dq

exp {-[V(q) + p2

24

2V(q)]} (34)

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which has been extensively used in molecular dynamicssimulations of liquids.18,33

The values of qvQFH and qv

QFH for H2 are shown in Table 4and Figure 1, respectively. As shown, the behavior of QFHvibrational partition function is similar to the LCP one. The

basic difference occurs at low temperatures, where qvQFH

underestimates the exact value of the vibrational partition

function while qvLCP overestimates it. For example, at T) 700

K, qvQFH is 63% smaller than the quantum one, whereas qv

LCP is

larger than qvQM by about 242%. In general, the QFH ap-proximation leads to results better than the LCP approximation,reaching an accuracy of 10% at T 1400 K and converging tothe quantum sum-over-states for T 2400 K. However, as for

qvLCP, qv

QFH diverges for high temperatures.3.5. Pitzer-Gwinn Approximation. The scheme developed

by Pitzer and Gwinn41 has a semiempirical nature and wasoriginally developed to include quantum corrections in thecalculation of thermodynamic functions for molecules withinternal rotation. In their approach, the partition function isapproximated by its classical component scaled by the ratio ofthe quantum over classical partition functions for a referencesystem where both are known exactly. If the reference systemis described by a harmonic potential, then one obtains41-43

where qvCM(T) is the classical partition function, and

are, respectively, the quantum and classical partition functionsfor the harmonic oscillator calculated with the zero of energy

at the bottom of the potential curve; V is the vibrationalfundamental frequency. The zero point energy 0 appearing in

the expression for qHOQM can be calculated by using different

methods,42 such as the harmonic and Dunham formulas. In thiswork, we have used the relation 0 ) hV/2. Furthermore, sincethe zeroth and the first vibrational levels are known, we haveconsidered the frequency in eq 36 to be the frequencyassociated to the 1r0 transition. Despite its simple form, thePG correction is known to be fairly accurate and hence isfrequently used.63

Numerical values of qvPG and the associated Monte Carlo

errors are reported for H2 in Table 4, while qvPG is depicted in

Figure 1. For temperatures below 1900 K, they show a betteragreement with the quantum result than all previous corrections

(qvWK, qvGOR, qvLCP, and qvQFH). In fact, for temperatures of a fewhundred kelvin, qv

PG still gives values of the same order of

magnitude of the quantum ones (qvPG 0.50 at T) 300 K,

whereas qvPG reaches an accuracy of 10% at T 900 K). For

intermediate temperatures, the other approaches correct betterthe classical partition function: the QFH partition function isbetter than the PG one for 1900 e T/K e 5200, and the WKexpansion gives better results for 2900 e T/K e 4500. The

Pitzer-Gwinn values converge to qvQM for lower temperatures

values than in the GOR and LCP ones: T 3200 K for qvPG, T

6900 K for qvGOR, and T 4000 K for qv

LCP. Moreover, forthe highest temperature considered in the present work (T)

9000 K) a deviation of only 0.07% is still obtained for qvPG.

4. A Hybrid LCP/QFH Correction

We have found that qvWK and qv

GOR diverge at low tempera-tures because the corresponding expansions (probability distri-bution for WK and quantum Hamiltonian for GOR) in phasespace formalism are truncated at p2. However, they both con-verge to the quantum sum-over-states quicker than the classical

result. On the other hand, the LCP and QFH methods give closeresults to qvQM at low temperatures: qv

LCP overestimates whereas

qvQFH underestimates the quantum vibrational partition func-

tion. Unfortunately, both the methods diverge at high temper-atures since they are based on the solution of the configurationalintegral rather than the total phase space integral in eq 2. Inthis section, we suggest an hybrid scheme which aims to keepthe advantages of the LCP and QFH methods while discardingtheir nondesirable features.

From the GOR approach, it is easy to demonstrate25 byintegrating the phase space integral in eq 32 over the momentabetween- and that the effective quantum potential is givenby

where the p2 term can be seen as a mixture of the correctionterms in eqs 33 and 34. Moreover, we have seen in section 3.2that the GOR approach diverges at low temperatures, which ismainly due to the fact that -2(V(q))2 assumes very largenegative values. Conversely, the LCP method has been shownto reach large positive deviations at low temperatures. Thissuggests that an average of the LCP and QFH quantum effec-tive potentials may lead to acceptable results at such low-temperature regimes. Equivalently, the terms 2V(q) and2(V(q))2 may be thought as being of the same order ofmagnitude, and hence be profitably combined. Thus, our

qVPG

(T) ) qVCM

(T)qHO

QM(T)

qHOCM

(T)(35)

qHOQM

(T) )exp(-0)

1 - exp(-h)and qHO

CM(T) )

1

h

(36)

TABLE 5: Rovibrational Partition Functiona of HClCalculated Using Various Approaches

T/K qvrCM

qvrLCP/QFH

qvrQM

qvrFPIb

300 1.1690 3.7718(-3)c 1.6526(-2) 1.67(-2)1.7(-1)d 7.2(-4) 8.0(-4)

400 2.2370 7.0418(-2) 1.3008(-1) 1.39(-1)2.7(-1) 1.2(-2) 5(-3)

500 3.6074 3.4172(-1) 4.7205(-1) 4.9(-1)3.8(-2) 4.7 (-2) 2.0(-2)

600 5.2885 9.5499(-1) 1.1538 1.17

4.7(-1) 1.1(-1) 2.0(-2)900 12.248 5.4206 5.7147 5.76

7.6(-1) 3.8(-1) 7.0(-2)1000 15.217 7.8115 8.0999

8.4(-1) 4.8(-1)1200 22.139 13.851 14.115 14.2

1.0 6.9(-1) 2.0(-1)2000 63.324 53.240 53.569 53.6

1.7 1.5 5.0(-1)4000 268.19 257.04 257.62 257.0

3.5 3.4 2.05000 431.70 420.43 421.11 420.0

4.4 4.3 3.06000 641.27 629.99 630.74 636.0

5.3 5.3 5.07000 900.13 888.91 889.65

6.2 6.29000 1567.8 1557.0 1557.38.1 8.0

a See the text for nomenclature. b Fourier path integral (FPI) resultsfrom ref 64. c Given in parantheses is the power of 10 by which thenumbers should be multiplied. d Second entry for each temperature givesthe standard Monte Carlo deviation.

VQM(q)) V(q) + p2

24{22V(q) -2 (V(q))2} (37)



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approach consists of adding the following effective potentialto the classical Hamiltonian:

where

and implies the above-discussed restriction on the integrationhypervolume.

For the diatomic case, the effective potential reduces to

From eq 38, one then obtains the rovibrational partition functionas

while the vibrational partition function assumes the form

The results of such an hybrid approach to the vibrationalpartition function are presented in Table 4 for the vibrationalpartition function of H2. Clearly, for low and medium temper-atures, the agreement of q

vLCP/QFH with the quantum results is

good. Moreover, no sign of divergence is manifest at hightemperatures which contrasts with the LCP and QFH ap-proaches. In fact, q

vLCP/QFH reaches an accuracy of 10% at T =

750 K while starting to converge to qvQM at T = 2100 K. In

addition, at T) 9000 K, a deviation of only 0.03% is obtained.

Although qvLCP/QFH gives the best overall results when com-

pared with the quantum ones, the qvPG

and qvQFH

methods arestill the best for T e 700 K and 2600 e T/K e 4800,respectively. These features can also be seen from Figure 1.

To test further the validity of the LCP/QFH approachsuggested in the present work, we have computed the rovibra-tional partition function of HCl and ArO by using eq 41. Theresults are given in Table 5 and Table 6. For comparison, wealso give the exact quantum and classical results from eq 1 andeq 2. For comparison, we also report the results obtained by

using the Fourier path integral formalism64 (qvrFPI

) based on thesame potential curve.

Table 5 shows that qvrLCP/QFH is much more accurate than qvr

CM

over the whole range of temperatures. In comparison with qvrQM

and qvrFPI, the agreement is seen to be satisfactory even at low

temperatures. Specifically, at T ) 600 K, where qvrCM has anerror of 358% and the Fourier path integral formalism has an

error of 1.4%, one finds for qvrLCP/QFH a deviation of 17%. At

medium temperatures (T 2000 K) qvrLCP/QFH is comparable in

accuracy to qvrFPI, while becaming more accurate with increas-

ing temperature.As expected,13 Table 6 shows that similar considerations apply

for ArO but for much lower temperatures than in the cases of

TABLE 6: Rovibrational Partition Functiona of ArOCalculated Using Various Approaches

T/K qVrCM

qVrLCP/QFH

qVrQM

2.5 8.0560(-1)b 1.6874(-4) 2.6069(-3)3.8(-2)c 1.6(-5)

5.0 3.2648 3.0101(-1) 4.0479(-1)7.8(-2) 9.7(-3)

10.0 13.956 7.6740 7.47531.6(-1) 9.1(-2)

15.0 33.432 25.973 25.254

2.3(-1) 1.8(-1)25.0 93.008 85.760 83.976

3.7(-1) 3.4(-1)50.0 256.73 252.65 249.04

6.7(-1) 6.6(-1)75.0 377.50 375.15 371.48

8.7(-1) 8.7(-1)100.0 461.48 459.99 456.78

1.0 1.0

a See the text for nomenclature. b Given in parantheses is the powerof 10 by which the numbers should be multiplied. c Second entry foreach temperature gives the standard Monte Carlo deviation.

TABLE 7: Ratio of Various Partition Functionsa as a Function of the Reduced Temperature for H2, HCl, and ArO

H2 HCl ArO

T/K qvCM

/qvQM

qvLCP/QFH

/qvQM

T/K qvrCM

/qvrQM

qvrLCP/QFH

/qvrQM

T/K qvrCM

/qvrQM

qvrLCP/QFH

/qvrQM

0.2 623.2 14.763 0.7738 427.0 13.255 0.6012 4.36 13.266 0.61140.3 934.8 4.1841 0.9733 640.4 3.9282 0.8557 6.54 3.8321 0.92570.4 1246.5 2.4141 1.0075 853.9 2.3090 0.9386 8.72 2.2414 1.01250.5 1558.1 1.8100 1.0126 1067.4 1.7500 0.9718 10.90 1.6977 1.03020.6 1869.7 1.5291 1.0118 1280.9 1.4889 0.9848 13.08 1.4471 1.03070.7 2181.3 1.3742 1.0100 1494.3 1.3450 0.9899 15.26 1.3114 1.02820.8 2492.9 1.2791 1.0081 1707.8 1.2568 0.9922 17.44 1.2302 1.02570.9 2804.6 1.2164 1.0066 1921.3 1.1986 0.9935 19.61 1.1779 1.02391.0 3116.2 1.1726 1.0053 2134.8 1.1582 0.9944 21.79 1.1423 1.02261.2 3739.4 1.1171 1.0035 2561.7 1.1070 0.9957 26.15 1.0984 1.02091.4 4362.6 1.0844 1.0023 2988.7 1.0769 0.9966 30.51 1.0732 1.01961.6 4985.9 1.0634 1.0015 3415.7 1.0578 0.9972 34.87 1.0573 1.01841.8 5609.1 1.0492 1.0009 3842.6 1.0448 0.9976 39.23 1.0465 1.01722.0 6232.4 1.0392 1.0005 4269.6 1.0356 0.9980 43.59 1.0388 1.01612.5 7790.4 1.0241 1.0002 5337.0 1.0217 0.9985 54.49 1.0269 1.0135

a See the text for nomenclature.

qVr

LCP/QFH(T) )

1

hn dqdp exp {-[H

CM(q, p) +

Veff(q)]} (38)

Veff

(q) )p

2

48{

2V(q) +(V(q))

2} (39)

Veff

(r) )p

2

48{( d2

dr2+

1

r

d

dr)V(r) + ( ddrV(r))2} (40)

qVr

LCP/QFH(T) )

1

h3 drd d dprdp dp

exp{-[pr2

2+

p2

2r2+

p2

2r2 sin2+ V(r) + V

eff(r)]} (41)

qV

LCP/QFH(T) )

1

h drdpr

exp {-[pr2

2+ V(r) + V

eff(r)]} (42)

5278 J. Phys. Chem. A, Vol. 105, No. 21, 2001 Prudente et al.


8/8

H2 and HCl. For example, at T) 5.0 K where qvrCM shows a

deviation of 700%, qvrLCP/QFH has an error of only 25.6%. Such

an observation can be rationalized from the zero point of energy(0) values of the three molecules: 0 ) 2165.866 cm-1 forH2, 0 ) 1483.759 cm-1 for HCl, and 0 ) 15.148 cm-1 forArO. For this, we use the following reduced temperature13

where T is in K, and 0 in cm-1. Table 7 reports the values of

the ratios qvrCM/qvr

QM and qvrLCP/QFH/qvr

QM as a function of ; forreference, the corresponding Tvalues are also tabulated. Clearly,similar trends are observed for the deviations of qLCP/QFH fromqQM as a function of . For example, at ) 0.5, qCM shows adeviation of 81% for H2, 75% for HCl, and 70% for ArO, whilethe corresponding deviations for qLCP/QFH are 1.3%, 2.8%, and3.0%. Moreover, for = 1.2, qCM shows an error of 10% forthe three diatomic systems while qLCP/QFH reaches the sameaccuracy at = 0.25 for H2 and ArO, and at = 0.35 for HCl.Thus, the LCP/QFH hybrid method proposed here can reliablybe used to calculate the rovibrational partition functions of

diatomic molecules for temperatures above T) 0.30/kB.

5. Conclusions

Several quantum, semiclassical, and semiempirical correctionsto the classical partition function of molecular systems havebeen analized. In addition, a novel hybrid procedure (LCP/QFH)has been proposed. A numerical application to H2 has shownthat this new method performs generally better than previousones, namely for temperatures above 700 K. Similar perfor-mances are anticipated for the HCl and ArO cases also reportedin this work. In fact, it blends the advantages of two popularsemiclassical methods (LCP and QFH) but restricted to thebound phase space. Although applications were done only fordiatomic molecules, there is no reason of principle why it should

not work equally well for larger polyatomics. Work along theselines is currently in progress.

Acknowledgment. This work has the support of Fundacaopara a Ciencia e Tecnologia, Portugal, under program PRAXISXXI. It has also benefited from an EC grant under ContractCHRX-CT 94-0436. One of us (F.V.P.) also acknowledgespartial financial support from Fundacao Coordenacao de Aper-feicoamento de Pessoal de Nvel Superior (CAPES, Brazil).

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)kBT

0) 0.695

T

0(43)


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Frederico V. Prudente, Antonio Riganelli and Antonio J. C. Varandas- Calculation of the Rovibrational Partition Function Using Classical Methods with Quantum Corrections